The order-theory tag has no wiki summary.

**17**

votes

**2**answers

675 views

### Which are the rigid suborders of the real line?

Which are the rigid suborders of the real line?
If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ...

**2**

votes

**3**answers

503 views

### Countable atomless boolean algebra covered by a larger boolean algebra

Suppose Q is an atomless countable boolean algebra, and B is an arbitrary atomless boolean algebra. Q is unique modulo isomorphisms. There is a subalgebra in B that is isomorphic to Q. There is ...

**6**

votes

**2**answers

965 views

### Ordinals that are not sets

The class of all ordinal numbers $\mathbf{Ord}$, aside being a proper class, can be thought of an ordinal number (of course it contains all ordinal numbers that are sets, not itself). Then one could ...

**20**

votes

**2**answers

669 views

### How much choice is needed to show that formally real fields can be ordered?

Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ...

**3**

votes

**2**answers

318 views

### Semilattices in atomless boolean algebras

Let S be a bounded semilattice without maximal elements. Can we always construct an atomless boolean algebra B, containing S as a subsemilattice, such that S is cofinal in B-{1}? That is, for every ...

**0**

votes

**2**answers

300 views

### name for “solid” subset of a partially ordered set?

For P a partially ordered set, let S be a subset of P such that if:
a,c\in S and b\in P and a<=b<=c then b\in S
Is there a name for a subset with this property? The term "dense" subset is ...

**-3**

votes

**1**answer

224 views

### Ordering of tuples equivalent to mapping to R?

Suppose we have a non-strict total ordering on tuples of real numbers (the ordering includes tuples of differing lengths). Any tuple, t2, generated from another tuple, t1, by increasing one or more ...

**4**

votes

**2**answers

248 views

### Is there a poset with 0 with countable automorphism group?

Is there a poset P with a unique least element, such that every element is covered by finitely many other elements of P (and P is locally finite -- actually, per David Speyer's example, let's say that ...

**0**

votes

**1**answer

296 views

### Determining set membership from ordering relationships among disjoint sets

Suppose we have a set $P$ (an infinite set), and we have a partition of $P$ into (finitely many) disjoint subsets $P_i$, so that $P = \cup_i P_i$, and $P_i \cap P_j = \emptyset$ for $i \neq j$.
...

**3**

votes

**2**answers

397 views

### Unbounded countable subset

(Edit: The first formulation is wrong. See the second answer) Does every totally ordered set contain an unbounded countable subset. In other words: If S is a totally ordered set, can we find a (edit: ...

**8**

votes

**6**answers

2k views

### Generalizations of Boolean posets/lattices

A Boolean lattice has a number of rather nice properties which give it a central role in many parts of combinatorics. For instance, it's a lattice, it can be augmented with a ring structure, it can ...

**19**

votes

**6**answers

2k views

### What's a non-abelian totally ordered group?

Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a total ordering (not to be confused with a well-ordering) ...

**2**

votes

**2**answers

388 views

### Automorphisms of the totally ordered group Z^n with lexicographical order

It is easy to see that the totally ordered group Z (the integers) with the natural order has no non-trivial automorphisms. Is this also true for Z^n with the lexicographical order?

**2**

votes

**1**answer

427 views

### Chains intersecting antichains in finite posets

I feel a little embarrassed to be asking this question here, since I think it should be much easier than I'm making it, but here goes:
Given a finite poset P, does there necessarily exist some chain ...

**3**

votes

**3**answers

534 views

### Is there a “universal LYM inequality?”

This question is based on a blog post of Qiaochu Yuan.
Let P be a locally finite* graded poset with a minimal element, and w be a weight function on the elements of P. Suppose that the total weight ...

**10**

votes

**5**answers

1k views

### Questions about ordering of reals and irrationals

Three problems from G.Rosenstein "Linear orderings" (from the end of Chapter 2 and beginning of Chapter 4):
1) Is there a nondecreasing function from irrationals onto reals?
2) Is there a ...